I really like OP's approach here but still, find that some kids' specific games are quite good on their own. In particular, Ticket to Ride: First Journey is a great adaptation to the full game.
Another one that nobody believes me about (because it sounds like such a preposterous cash grab) is Super Mario Brothers Monopoly. But the changed some of the fundamentally boring aspects of the game to make it really quick to play and introduced the concept of "hero" characters with unique abilities to it.
The game I've found with the most cross-generational appeal is Kingdomino. I've played it with 3 year olds and teenagers at the same time, and both enjoyed it. They only change we make to the rules for the younger kids is removing the 5x5 restriction on the size of your kingdom.
Our family has also found this to be true for Catan Junior. It's a great game on its own with quite a lot of replay value, so we end up sometimes playing Catan Junior even though everyone has the ability to play Settlers of Catan.
I also like this approach. And I've found the opposite way around also works pretty well: kids (say age 4-6+) are perfectly capable of playing the full adult versions of games.
I've found the challenge isn't so much for them to understand the rules as much is it's holding their attention with something not as instantly gratifying as the other entertainment they are used to in apps and video streaming.
Ah, my siblings and I were only allowed an hour of screen time a week when we younger. So that wasn't really an issue. I think an hour a week is a bit extreme, but I definitely intend to put strong limits on my children, especially when they're young.
My oldest kid once mopped the floor with me, 7 games in a row at Love Letter. I didn't throw the games to make him happy, he just plain had a combination of good draws and good play. He was 7 at the time. Kids are amazing.
Indeed, there are also plenty of great board games that are targeted towards kids! Don't need to run simplified Adult games.
The HABA publishing series is always a hit. Rhino Hero working for kids and adults alike.
First Orchard teaches the basics of game mechanics to younger kids.
Set Junior simplifies Set for kiddos.
Ice Cool is a great dexterity game.
The Magic Labyrinth (2009) involves mental mapping and memorization skills.
Looney Quest is a great drawing game as you try to collect coins and avoid the bad stuff.
Not to mention all the Junior games of heavy classics like Ticket to Ride: First Journey, My First Stone Age, etc.
Obviously the downside is they likely will outgrow these simplified versions. So the value of hacking the games you love as an adult is definitely a big plus and lets you save money and work them towards great board games. But on the other hand, why strip out all the pieces when you can play something designed for that age demographic?
The one major improvement is that you're having to buy just the full game instead of 2 variants. The chances are you already have the original full game, and using that to teach a reduced set makes sense if you don't wanna (or can't) buy the smaller set because it will stop being fun in a year once the kid learns the main game.
There is also the fact that some games won't even have reduced sets (Guillotine for eg), or will have it but it will be impossibly hard to get where I live. I can easily get a copy of TTR in India, but finding a copy of TTR:1st Journey is much tougher.
Target (the stores) has been a really interesting force for encouraging board game designers to adapt their games to entry level. TtR: First Journey, Catan Junior, Evolution's base game rebranded and cleaned up to be an entry level game leaving Evolution: Climate the new "base game" for more experienced groups, Captain Sonar Junior. So far it seems that most designers that have accepted Target's challenge have built great adaptations that don't water down their main concepts too much and are great starting points to the deeper games.
The best Monopoly is the one on the NES with a four-score adapter :-)
All the standard rules, no-one can modify them, runs the auctions for you, plays very fast. D-pad and four buttons (counting select and start) so even young kids (or oldsters) with no gaming experience can pick it up fast. I wouldn't ordinarily recommend a video game as a complete replacement for the physical version of a board game, but in the case of Monopoly, yes, definitely, find the simplest video game version you can and use that.
We've had really good success with Magic Maze Kids - as well as Ticket to Ride: First Journey and Catan Junior (both of which are fantastic). The thing that Magic Maze Kids has going for it is an _absolutely fabulous_ tutorial system that slowly introduces every mechanic of the game.
Other games like Pigasus and Snowman Dice (both by Brain Games I think) have been a hit too.
I think if you're willing to be VERY flexible with all rules, 2.5 is old enough to play all of the above.
When we play with younger kids we try really hard to emphasize fun and downplay stringent rule following, for better or for worse.
Some games, like snowman dice, are fun just for the sake of it. It's fun to roll the dice and push them around.
I'd say Catan Junior is fine for a 3 year old, but make sure to spend the first few times just familiarizing the rules (i.e. making the connection between getting a resource and spending it by matching up the resources on your resource card).
Ticket to Ride Junior is a little bit harder, but a 3 year old should do ok with help. Again, if you really "take it easy" about the rules a very young child can have a lot of fun with these if for nothing other than the colors and manipulatives/meeples alone.
Magic Maze Kids we did starting at 2.5-3ish and it was fine.
Pigasus, on the other hand, a 2 year old can do and really enjoy. That game is absolutely amazing because it's one of those things that a small child can do better than some adults (like... for real!). Also the sound the pig makes when you squeeze it is incredibly funny to most kids and very rewarding. It's a fun one.
My 3yo daughter can handle Ticket to Ride First Journey, and even now knows some of the city names (we actually have a German edition, since I couldn't find an English one at a sensible price). It plays a bit quicker than the full version, so tends to hold her attention longer.
I've played Carcassonne with kids who couldn't read yet; it's great! No language dependence whatsoever!
OTOH, I typically teach adults how to play Race for the Galaxy by starting out with a few hands of San Juan... It's a very similar, but much much simpler game.
Another one that nobody believes me about (because it sounds like such a preposterous cash grab) is Super Mario Brothers Monopoly. But the changed some of the fundamentally boring aspects of the game to make it really quick to play and introduced the concept of "hero" characters with unique abilities to it.